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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Substitute (<a href="" class="xref" data-knowl="./knowl/eq8_6.html" title="Equation 6.6.3">(6.6.3)</a>) into (<a href="" class="xref" data-knowl="./knowl/eq8_4.html" title="Equation 6.6.1">(6.6.1)</a>):</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_6.html ./knowl/eq8_4.html ./knowl/eq8_7.html ./knowl/eq8_8.html ./knowl/eq8_1.html">
\begin{equation}
{\bm \Psi}^{\prime}(t)\, {\bf u}(t)+{\bm \Psi}(t)\, {\bf u}^{\prime}(t)={\bf P}(t) {\bm \Psi}(t) {\bf u}(t)+{\bf g}(t),\tag{6.6.4}
\end{equation}
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<p class="continuation">where</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_6.html ./knowl/eq8_4.html ./knowl/eq8_7.html ./knowl/eq8_8.html ./knowl/eq8_1.html">
\begin{equation}
{\bm \Psi}^{\prime}(t)=\left[{\bf x}^{\prime (1)}, {\bf x}^{\prime (2)},\cdots, {\bf x}^{\prime (n)}\right]
=\left[{\bf P}(t)\, {\bf x}^{(1)}, {\bf P}(t)\, {\bf x}^{(2)}, \cdots, {\bf P}(t)\, {\bf x}^{(n)}\right]
={\bf P}(t) \left[{\bf x}^{(1)}, {\bf x}^{(2)}, \cdots, {\bf x}^{(n)}\right]
={\bf P}(t) \, {\bm \Psi}(t).\tag{6.6.5}
\end{equation}
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<p class="continuation">Based on (<a href="" class="xref" data-knowl="./knowl/eq8_7.html" title="Equation 6.6.4">(6.6.4)</a>) and (<a href="" class="xref" data-knowl="./knowl/eq8_8.html" title="Equation 6.6.5">(6.6.5)</a>)</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_6.html ./knowl/eq8_4.html ./knowl/eq8_7.html ./knowl/eq8_8.html ./knowl/eq8_1.html">
\begin{equation*}
{\bm \Psi}(t)\, {\bf u}^{\prime}(t)={\bf g}(t).
\end{equation*}
</div>
<p class="continuation">Multiplying <span class="process-math">\({\bm \Psi}^{-1}(t)\text{,}\)</span> we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_6.html ./knowl/eq8_4.html ./knowl/eq8_7.html ./knowl/eq8_8.html ./knowl/eq8_1.html">
\begin{equation*}
{\bf u}^{\prime}(t)={\bm \Psi}^{-1}(t)\,{\bf g}(t) \to 
{\bf u}(t)=\int {\bm \Psi}^{-1}(t)\,{\bf g}(t) \mathrm{d} t+k_1,
\end{equation*}
</div>
<p class="continuation">where we may choose <span class="process-math">\(k_1=0\text{.}\)</span> The general solution for (<a href="" class="xref" data-knowl="./knowl/eq8_1.html" title="Equation 6.5.1">(6.5.1)</a>) is then</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_6.html ./knowl/eq8_4.html ./knowl/eq8_7.html ./knowl/eq8_8.html ./knowl/eq8_1.html">
\begin{equation*}
{\bf x}={\bm \Psi}(t)\, {\bf c}+{\bm \Psi}(t)\, \int {\bm \Psi}^{-1}(t)\,{\bf g}(t) \mathrm{d} t.
\end{equation*}
</div>
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